Geoffrey Hellman* and Stewart Shapiro.**Varieties of Continua—From Regions to Points and Back

Abstract

In this ambitious study, Hellman and Shapiro develop formal treatments of a regions-based approach to the mathematical continuum that rival the standard point-theoretic account, and then present a comparative study of the advantages and disadvantages of several main approaches to defining continuity. There are two main contrasts that they see between the regions-based or ‘point-free’ approach typical of the Aristotelian tradition, and the mainstream Dedekind–Cantor account of continuity. The first is that in the latter the continuum is construed as composed of points, whereas in the former it is composed of regions, not points. The second is that the latter is based on extensive appeal to the actual infinite, whereas the Aristotelian account eschews actual infinities in mathematics and works only with the potential infinite. Recourse to the potential infinite also underlies the constructivist approaches of L.E.J. Brouwer, Errett Bishop, and others, and the smooth infinitesimal analysis of Kock and Lawvere (as developed in particular by John L. Bell) depends on an intuitionistic logic to give credibility to the idea of rigorous accounts of the continuum that can rival the Dedekind–Cantor one. In contrast, Hellman and Shapiro seek to provide a ‘thorough-going non-punctiform conception of continua’ (p. vii), a regions-based account that eschews points as parts of regions, but without using intuitionistic logic.